Coulomb potential in one dimension with minimal length: A path integral approach

TL;DR

This paper derives exact energy eigenvalues and eigenfunctions for a one-dimensional Coulomb potential within a minimal length quantum framework using a path integral approach in momentum space.

Contribution

It introduces a novel path integral method in momentum space to solve the Coulomb problem with minimal length, providing exact spectral solutions.

Findings

Exact energy eigenvalues obtained

Momentum space eigenfunctions derived

Framework incorporates minimal length effects

Abstract

We solve the path integral in momentum space for a particle in the field of the Coulomb potential in one dimension in the framework of quantum mechanics with the minimal length given by $(\Delta X)_{0}=\hbar \sqrt{\beta}$, where $\beta$ is a small positive parameter. From the spectral decomposition of the fixed energy transition amplitude we obtain the exact energy eigenvalues and momentum space eigenfunctions.